Evaluate Log Expressions: Step-by-Step for Students

The study of logarithms, a cornerstone in mathematics education, often presents a challenge for students, particularly when dealing with log expressions. Khan Academy, a valuable resource for online learning, offers numerous tutorials, but sometimes a more structured approach is needed to truly grasp the concepts. Logarithmic scales, frequently used in fields like chemistry to measure pH levels, demonstrate the practical application of these mathematical tools. Understanding how to evaluate log expressions is essential for success in higher-level math courses and standardized tests such as the SAT, and it begins with mastering a step-by-step methodology.

Welcome to the intriguing realm of logarithms! Often perceived as a daunting mathematical concept, logarithms are, in reality, a powerful and versatile tool. This section aims to gently introduce you to their fundamental definition and purpose, demystifying the concept and sparking your curiosity.

Think of this as your friendly guide, making logarithms accessible and understandable, even if you're just starting your mathematical journey.

Defining Logarithms: Unveiling the Mystery

At its heart, a logarithm is simply the exponent to which a base must be raised to produce a given number.

Let's break that down. If we have an equation like b^y = x, the logarithm answers the question: "To what power (y) must we raise the base (b) to get x?". This is written as log_b(x) = y.

For example, log₂(8) = 3 because 2³ = 8. In this case, the logarithm tells us that we need to raise 2 to the power of 3 to get 8.

Logarithms as Inverse Operations: Reversing Exponents

Understanding logarithms becomes much easier when you grasp their relationship with exponents. Logarithms and exponents are inverse operations, meaning they "undo" each other.

Just like addition and subtraction, or multiplication and division, logarithms and exponents work in opposite directions.

If b^y = x, then log_b(x) = y. Notice how the logarithm essentially "isolates" the exponent. This ability to unravel exponential relationships is what makes logarithms so valuable.

Why Logarithms Matter: Real-World Applications

Logarithms aren't just abstract mathematical concepts; they're essential tools in various fields. They allow us to work with and understand very small and very large numbers on a scale that makes sense to humans.

Here are a few key examples:

• The Richter Scale: Used to measure the magnitude of earthquakes. Because earthquake intensity can vary so wildly, logarithms let us represent these measurements in a more manageable form.

• Sound Intensity (Decibels): The loudness of sound is measured in decibels, which are based on a logarithmic scale. This reflects how humans perceive sound; we don't experience sound intensity linearly.

• Chemical pH Levels: The pH of a solution, which indicates its acidity or alkalinity, is also measured using a logarithmic scale. Each pH unit represents a tenfold change in acidity.

• Computer Science: Logarithms are useful for analyzing the complexity of algorithms, particularly how efficient a search or sorting algorithm is.

These are just a few examples, but they demonstrate the pervasive nature and usefulness of logarithms.

Key Components: Understanding the Language of Logarithms

To confidently work with logarithms, you need to understand the key components of a logarithmic expression: the base and the argument.

Base of a Logarithm

The base (b) is the number that is raised to a power. It's a crucial part of the logarithm because it determines the scale.

In the expression log_b(x), 'b' is the base. The base is usually a positive number (other than 1).

Argument of a Logarithm

The argument (x) is the value you're taking the logarithm of. In the expression log_b(x), 'x' is the argument. The argument must always be a positive number.

Understanding the roles of the base and argument is fundamental to understanding how logarithms work. These two components, combined with the logarithm itself, allow us to express and solve a wide range of mathematical and scientific problems.

Revisiting Essential Mathematical Foundations

Welcome to the foundation upon which our logarithmic journey will be built! Logarithms rely heavily on concepts you may have already encountered in your mathematical studies. This section serves as a crucial refresher, ensuring you have a solid grasp of exponents, exponential functions, and the indispensable order of operations. Think of it as fortifying the groundwork before constructing a magnificent building.

The Power of Exponents: Unveiling the Basics

Exponents are the shorthand we use to express repeated multiplication. They tell us how many times a base number is multiplied by itself. For example, in the expression 2³, 2 is the base, and 3 is the exponent. This means 2 2 2, which equals 8.

Understanding exponential notation is crucial. Remember that any number raised to the power of 0 equals 1 (except 0 itself, which is undefined). And a number raised to the power of 1 is simply the number itself.

Mastering Exponential Operations

When working with exponents, several key rules simplify calculations:

• Product of Powers: When multiplying numbers with the same base, add the exponents. (x^m

  **xⁿ = x^m+n)

• Quotient of Powers: When dividing numbers with the same base, subtract the exponents. (x^m / xⁿ = x^m-n)

• Power of a Power: When raising a power to another power, multiply the exponents. ((x^m)ⁿ = x^m**n)

Familiarizing yourself with these rules will make navigating logarithmic expressions significantly easier.

Exponential Functions: Growth and Decay Unveiled

Exponential functions take the form f(x) = a^x, where a is a constant base (usually greater than 0 and not equal to 1) and x is the variable in the exponent.

These functions are characterized by their rapid growth or decay. If a is greater than 1, the function exhibits exponential growth. As x increases, the value of f(x) increases at an accelerating rate. Think of a rapidly growing investment.

Conversely, if a is between 0 and 1, the function exhibits exponential decay. As x increases, the value of f(x) decreases, approaching zero. A classic example is the radioactive decay of a substance.

The graphs of exponential functions are visually distinctive. Exponential growth curves sweep upwards, while exponential decay curves descend towards the x-axis. Understanding this visual representation is invaluable.

Order of Operations is King (PEMDAS/BODMAS)

In mathematics, the order of operations is a non-negotiable rule. It dictates the sequence in which operations must be performed to arrive at the correct answer. Remember the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction).

Multiplication and division are performed from left to right, as are addition and subtraction. When evaluating logarithmic expressions, adhering to this order is paramount.

The Perils of Ignoring Order

Forgetting or misapplying the order of operations can lead to drastically wrong answers. For example, consider the expression 2 + 3 4. If we incorrectly add first, we get 5 4 = 20. However, the correct answer, following PEMDAS/BODMAS, is 2 + (3 * 4) = 2 + 12 = 14.

Always double-check the order of operations to ensure accuracy.

Solving Equations with Logarithms: A Sneak Peek

While we will delve deeper into solving equations later, it's important to acknowledge how logarithms come into play. Logarithms are particularly useful when solving exponential equations where the variable is in the exponent.

For instance, consider the equation 2^x = 8. We can use logarithms to isolate x. This involves taking the logarithm of both sides of the equation.

This is just a brief glimpse of the power of logarithms in solving equations. As we progress, you'll gain a comprehensive understanding of this technique.

By revisiting these fundamental concepts, we have established a strong foundation for our exploration of logarithms. With exponents, exponential functions, and the order of operations firmly in mind, we are now ready to unlock the secrets of logarithms.

Types of Logarithms: Common and Natural

Having armed ourselves with the fundamental understanding of logarithms and their intimate dance with exponents, we are now ready to explore the two most prominent figures in the logarithmic landscape: the common logarithm and the natural logarithm. These aren't just arbitrary flavors of logarithms; they are workhorses, frequently encountered in various scientific, engineering, and mathematical applications. Let's dive in and understand what makes them special.

The Common Logarithm (Base 10)

The common logarithm, as its name suggests, is the logarithm with a base of 10. It's the logarithm most people think of when they hear the word "logarithm," and you'll often find it on calculators as the "log" button.

Notation and Definition

The common logarithm of a number x is written as log₁₀(x). However, it is so frequently used that it's often abbreviated simply as log(x).

Essentially, log(x) answers the question: "To what power must we raise 10 to get x?"

In mathematical terms: If log(x) = y, then 10^y = x.

Evaluating Common Logarithms

Let's consider a few examples to solidify our understanding:

• log(100) = 2 because 10² = 100.

• log(1000) = 3 because 10³ = 1000.

• log(0.1) = -1 because 10^-1 = 0.1.

Notice the pattern? The common logarithm essentially counts the number of factors of 10 in a number.

Applications of the Common Logarithm

The common logarithm is instrumental in scenarios where quantities span several orders of magnitude.

• The Richter Scale: This scale, used to measure the magnitude of earthquakes, is a base-10 logarithmic scale. Each whole number increase on the Richter scale represents a tenfold increase in the amplitude of the seismic waves.

• Sound Intensity (Decibels): The decibel scale, used to measure sound intensity, is also a logarithmic scale based on the common logarithm. A small increase in decibels represents a significant increase in sound intensity.

The Natural Logarithm (Base e)

The natural logarithm is another extremely important type of logarithm. Instead of base 10, it uses the base e, where e is an irrational number approximately equal to 2.71828. This number, like pi, is a fundamental constant in mathematics.

Notation and Definition

The natural logarithm of a number x is written as ln(x). The "ln" is short for logarithmus naturalis, Latin for "natural logarithm."

The expression ln(x) answers the question: "To what power must we raise e to get x?"

Mathematically: If ln(x) = y, then e^y = x.

Understanding Euler's Number (e)

You might wonder why this seemingly strange number, e, is so important. It arises naturally in many areas of mathematics, particularly in calculus and exponential growth/decay models. It's the base of choice for continuous growth processes.

Evaluating Natural Logarithms

Let's look at some examples of evaluating natural logarithms:

• ln(e) = 1 because e¹ = e.

• ln(1) = 0 because e⁰ = 1.

• ln(e²) = 2 because e² = e².

Applications of the Natural Logarithm

The natural logarithm appears ubiquitously in science and engineering.

• Exponential Growth and Decay: The natural logarithm is essential in modeling processes that exhibit exponential growth or decay, such as population growth, radioactive decay, and compound interest.

• Calculus: The natural logarithm has particularly nice properties in calculus, making it a favorite for derivatives and integrals.

• Physics and Engineering: The natural logarithm is used in various physics and engineering applications, including thermodynamics, electrical circuits, and signal processing.

In summary, both the common logarithm and the natural logarithm are powerful tools with distinct applications. Understanding their definitions and properties is crucial for anyone venturing further into mathematics, science, or engineering.

Unlocking the Power: Properties of Logarithms

Having armed ourselves with the fundamental understanding of logarithms and their intimate dance with exponents, we are now ready to explore the properties of logarithms.

These aren't just abstract mathematical curiosities; they are powerful tools that allow us to manipulate, simplify, and solve logarithmic expressions with elegance and efficiency. Mastering these properties is key to unlocking the full potential of logarithms.

The Product Rule: Unveiling Multiplication's Secret

The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is expressed as:

log_b(xy) = log_b(x) + log_b(y)

In simpler terms, if you're taking the logarithm of two numbers multiplied together, you can separate them into two separate logarithms that are added together.

This rule is immensely helpful for breaking down complex logarithmic expressions.

Example: Simplifying with the Product Rule

Let's simplify the expression log₂(8

**4).

Using the product rule, we can rewrite this as:

log₂(8) + log₂(4)

Now, we know that log₂(8) = 3 and log₂(4) = 2, so:

3 + 2 = 5

Therefore, log₂(8** 4) = 5. We can verify this directly: log₂(32) = 5.

The Quotient Rule: Division Deconstructed

The quotient rule mirrors the product rule but applies to division. It states that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator.

Expressed mathematically:

log_b(x/y) = log_b(x) - log_b(y)

Think of it this way: when dividing inside a logarithm, you can split it into the subtraction of two logarithms.

Example: Applying the Quotient Rule

Let's simplify the expression log₃(81/3).

Applying the quotient rule, we get:

log₃(81) - log₃(3)

We know that log₃(81) = 4 and log₃(3) = 1, so:

4 - 1 = 3

Therefore, log₃(81/3) = 3. Again, we can confirm: log₃(27) = 3.

The Power Rule: Taming Exponents

The power rule is particularly useful when dealing with exponents within logarithms. It states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number.

The formula is:

log_b(x^p) = p

**log_b(x)

This means you can take the exponent and bring it down as a coefficient in front of the logarithm.

Example: Leveraging the Power Rule

Let's simplify log₅(25³).

Using the power rule, we can rewrite this as:

3** log₅(25)

Since log₅(25) = 2, we have:

3 * 2 = 6

Therefore, log₅(25³) = 6.

The Change of Base Rule: Adapting to Different Bases

The change of base rule is essential for evaluating logarithms with bases that are not directly supported by your calculator (which typically only handles base 10 and base e).

It allows you to convert a logarithm from one base to another.

The rule states:

log_a(x) = log_b(x) / log_b(a)

Where a is the original base, x is the argument, and b is the new base (usually 10 or e).

Using the Change of Base Formula

This rule is invaluable when using a calculator to evaluate logarithms with bases other than 10 or e. Simply convert to base 10 or base e and use your calculator's log or ln function.

Example: Changing the Base

Let's evaluate log₇(49). Most calculators don't have a log base 7 function. We can use the change of base rule to convert this to base 10:

log₇(49) = log₁₀(49) / log₁₀(7)

Using a calculator, we find that log₁₀(49) ≈ 1.690 and log₁₀(7) ≈ 0.845. Therefore:

log₇(49) ≈ 1.690 / 0.845 ≈ 2

Which confirms that log₇(49) = 2.

By mastering these logarithmic properties, you'll gain the ability to simplify complex expressions, solve challenging equations, and unlock a deeper understanding of the mathematical world. Remember to practice and apply these rules consistently to solidify your understanding.

Simplification and Evaluation: Mastering the Techniques

Having armed ourselves with the fundamental understanding of logarithms and their intimate dance with exponents, we are now ready to explore the properties of logarithms.

These aren't just abstract mathematical curiosities; they are powerful tools that allow us to manipulate, simplify, and solve logarithmic expressions with greater ease.

This section is your practical guide to mastering these techniques. We will delve into the step-by-step simplification of logarithmic expressions and learn how to evaluate logarithms using calculators effectively.

Step-by-Step Simplification: Unlocking Logarithmic Expressions

Simplifying logarithmic expressions might seem daunting at first, but with a systematic approach and a solid grasp of the properties we discussed, it becomes a manageable and even enjoyable process.

Remember, the key is to break down complex expressions into smaller, more digestible steps. Let's explore a couple of examples.

Example 1: Simplifying log₂(8x⁵/y²)

Let's tackle this expression using the properties of logarithms.

Step 1: Applying the Quotient Rule

First, recognize that we have a quotient inside the logarithm. We can use the quotient rule to separate the numerator and the denominator:

log₂(8x⁵/y²) = log₂(8x⁵) - log₂(y²)

Step 2: Applying the Product Rule

Now, let's focus on the first term, log₂(8x⁵). This involves a product, so we can apply the product rule:

log₂(8x⁵) = log₂(8) + log₂(x⁵)

Step 3: Applying the Power Rule

Next, we'll use the power rule to simplify log₂(x⁵):

log₂(x⁵) = 5

**log₂(x)

Step 4: Simplifying log₂(8) and log₂(y²)

We can simplify log₂(8) because 8 is a power of 2: log₂(8) = 3. Also, we can apply the power rule to log₂(y²): log₂(y²) = 2** log₂(y).

Step 5: Putting It All Together

Now, let's substitute everything back into our original expression:

log₂(8x⁵/y²) = 3 + 5 log₂(x) - 2 log₂(y)

Final Simplified Expression:

Therefore, the simplified expression is 3 + 5log₂(x) - 2log₂(y).

Example 2: Expanding ln(√(a³b)/c)

In this example, we will expand the given expression.

Step 1: Rewrite the square root

First, rewrite the square root as a fractional exponent:

ln(√(a³b)/c) = ln(((a³b)/c)^1/2)

Step 2: Apply the Power Rule

Apply the power rule to bring the exponent (1/2) outside the logarithm:

ln(((a³b)/c)^1/2) = (1/2)

**ln((a³b)/c)

Step 3: Apply the Quotient Rule

Next, we use the quotient rule to separate the numerator and the denominator:

(1/2) ln((a³b)/c) = (1/2) (ln(a³b) - ln(c))

Step 4: Apply the Product Rule

Apply the product rule to the term ln(a³b):

(1/2) (ln(a³b) - ln(c)) = (1/2) (ln(a³) + ln(b) - ln(c))

Step 5: Apply the Power Rule again

Apply the power rule to the term ln(a³):

(1/2) (ln(a³) + ln(b) - ln(c)) = (1/2) (3ln(a) + ln(b) - ln(c))

Step 6: Distribute (1/2) (Optional)

Distribute the (1/2) to each term (Optional):

(1/2)** (3ln(a) + ln(b) - ln(c)) = (3/2)ln(a) + (1/2)ln(b) - (1/2)ln(c)

Final Expanded Expression:

Therefore, the expanded expression is (3/2)ln(a) + (1/2)ln(b) - (1/2)ln(c).

Using Calculators for Evaluation: A Practical Guide

While understanding the properties of logarithms is crucial, sometimes we need to evaluate them numerically. That's where calculators come in handy.

Evaluating Logarithms with Standard Bases (10 and e)

Most calculators have dedicated buttons for common logarithms (log base 10) and natural logarithms (log base e, denoted as ln).

Simply enter the argument of the logarithm and press the corresponding button (log or ln). For example, to find log(50), enter "50" and press the "log" button.

Evaluating Logarithms with Non-Standard Bases

Many calculators don't have buttons for logarithms with arbitrary bases. This is where the change-of-base formula becomes invaluable. Remember the formula:

log_a(x) = log_b(x) / log_b(a)

To evaluate log₅(20) using a calculator, we can change the base to either 10 (log) or e (ln):

log₅(20) = log(20) / log(5) or log₅(20) = ln(20) / ln(5)

The calculator will then give the same approximate value, 1.861.

Tips for Accurate Calculations

• Use Parentheses Wisely: Ensure you use parentheses correctly, especially when dealing with complex expressions. For instance, log(2 + 3) is different from log(2) + 3.
• Double-Check Your Input: Mistakes can easily creep in when entering numbers into a calculator. Always double-check to ensure the input is correct.
• Be Mindful of the Order of Operations: Calculators follow the order of operations (PEMDAS/BODMAS). If you are unsure, break down the calculation into smaller steps.
• Use the Change-of-Base Formula Correctly: It is key that when using the change-of-base formula, you place the new based logarithm expression into parenthesis.

By mastering these simplification and evaluation techniques, you will be well-equipped to tackle a wide range of logarithmic problems and applications. Keep practicing, and soon you'll be navigating the world of logarithms with confidence!

Avoiding the Traps: Common Mistakes and How to Sidestep Them

Simplification and Evaluation: Mastering the Techniques Having armed ourselves with the fundamental understanding of logarithms and their intimate dance with exponents, we are now ready to explore the properties of logarithms. These aren't just abstract mathematical curiosities; they are powerful tools that allow us to manipulate, simplify, and sol...

Like navigating any intricate landscape, the world of logarithms presents its own set of pitfalls. Recognizing these common errors and equipping ourselves with strategies to avoid them is paramount for achieving accuracy and confidence in logarithmic calculations. Let's explore these pitfalls and equip you with the tools to navigate them successfully.

Identifying Common Logarithmic Mistakes

Mastering logarithms requires more than just memorizing formulas; it demands a deep understanding of the underlying concepts. Let's examine some of the most frequent errors that students encounter.

Misapplication of Logarithmic Properties

The properties of logarithms are powerful tools, but they must be applied correctly. A common mistake is incorrectly applying the product, quotient, or power rule. For example, students might mistakenly assume that log(x + y) is equal to log(x) + log(y), which is incorrect.

Remember, the properties apply to the logarithm of a product, quotient, or power, not the logarithm of a sum or difference. Always double-check that you are applying the rules to the correct operation within the logarithm.

Overlooking the Domain of Logarithmic Functions

One of the most critical aspects of logarithms that's often overlooked is their domain. The argument of a logarithm must always be positive. You can't take the logarithm of zero or a negative number.

For example, in the expression log(x - 2), x must be greater than 2. Failure to recognize this constraint can lead to incorrect or undefined solutions.

Always remember to check the domain before and after solving logarithmic equations to ensure your solutions are valid.

Confusing the Base and the Argument

The base and argument of a logarithm play distinct roles. Confusing them can lead to significant errors. For example, log₂(8) is not the same as log₈(2).

In log_b(x), 'b' is the base, and 'x' is the argument.

The base is the number that is raised to a power to obtain the argument. Understanding this fundamental relationship is crucial for accurate evaluation and simplification.

Strategies to Sidestep Logarithmic Mishaps

Now that we've identified the common pitfalls, let's arm ourselves with strategies to avoid them. Consistency and accuracy will be achieved through a thoughtful process.

Double-Checking Each Step

Mathematics is a sequential discipline; each step builds upon the previous one. Double-checking each step in your logarithmic calculations can catch errors early before they propagate through your entire solution.

Take a moment after each step to ensure you've applied the correct property, performed the arithmetic accurately, and maintained the correct notation. This simple habit can save you from significant frustration and wasted time.

Writing Out All Steps Clearly

While mental math can be tempting, especially as you become more confident, it can also be a source of errors.

Writing out all the steps in your logarithmic calculations, even the seemingly obvious ones, can significantly reduce the likelihood of mistakes.

A clear and organized presentation allows you to easily review your work, identify potential errors, and understand the logic behind each step. Treat each problem as a formal proof, where every step is justified.

Verifying Answers Using Estimations or Alternative Methods

Once you've arrived at a solution, don't just accept it at face value. Verify your answer using estimations or alternative methods.

For example, if you've solved an exponential equation using logarithms, plug your solution back into the original equation to see if it holds true. You can also use estimations to check if your answer is within a reasonable range. This provides an additional layer of security against calculation errors or misapplication of logarithmic properties.

By diligently applying these strategies, you can navigate the landscape of logarithms with greater confidence and accuracy. Remember, mathematics is not about avoiding mistakes, but learning from them. Embrace the challenge, and you'll find that logarithms become a powerful tool in your mathematical arsenal.

Practice Makes Perfect: Step-by-Step Examples

[Avoiding the Traps: Common Mistakes and How to Sidestep Them Simplification and Evaluation: Mastering the Techniques Having armed ourselves with the fundamental understanding of logarithms and their intimate dance with exponents, and having spent some time simplifying and evaluating these expressions, we are now ready to roll up our sleeves and work through some concrete examples. These aren't just abstract exercises; they are the building blocks to truly mastering the art of logarithms! So, grab your pencil and paper, and let's dive in together.]

Example 1: Solving an Exponential Equation Using Logarithms

Exponential equations, where the variable resides in the exponent, can seem daunting. Logarithms are our key to unlocking them.

Let's tackle the equation 5^x = 125.

Step 1: Recognize the Goal. Our aim is to isolate 'x'. Since it's in the exponent, we need a tool to bring it down. That tool is the logarithm.

Step 2: Apply the Logarithm. Take the logarithm of both sides of the equation. We can use any base, but the common logarithm (base 10) or the natural logarithm (base e) are often convenient for calculator use. Let's use the common logarithm here:

log(5^x) = log(125)

Step 3: Utilize the Power Rule. This is where the magic happens. The power rule of logarithms allows us to bring the exponent down as a coefficient:

x * log(5) = log(125)

Step 4: Isolate 'x'. Now, 'x' is simply multiplied by log(5). Divide both sides by log(5) to isolate it:

x = log(125) / log(5)

Step 5: Evaluate. Use your calculator to find the values of log(125) and log(5), then divide:

x ≈ 2.0969 / 0.6990 ≈ 3

Therefore, x = 3. (You can verify this by substituting x = 3 back into the original equation: 5³ = 125).

Example 2: Simplifying a Complex Logarithmic Expression

Now, let’s look at simplifying expressions using multiple log properties.

Consider this expression: log₂(16x³/y²).

Step 1: Apply the Quotient Rule. We see a division, so we can separate the logarithm into a difference:

log₂(16x³/y²) = log₂(16x³) - log₂(y²)

Step 2: Apply the Product Rule. Now, we have a product within the first logarithm, so we'll split that into a sum:

log₂(16x³) - log₂(y²) = log₂(16) + log₂(x³) - log₂(y²)

Step 3: Apply the Power Rule. We have exponents on 'x' and 'y', so we can bring those down as coefficients:

log₂(16) + log₂(x³) - log₂(y²) = log₂(16) + 3log₂(x) - 2log₂(y)

Step 4: Evaluate Known Logarithms. We know that 2⁴ = 16, so log₂(16) = 4:

4 + 3log₂(x) - 2log₂(y)

This is the simplified form of the expression.

Example 3: Evaluating a Logarithm with a Non-Standard Base

Sometimes, you'll encounter logarithms with bases that your calculator doesn't directly support. That's where the change of base formula comes to the rescue.

Let's say we need to find log₃(20).

Step 1: Apply the Change of Base Formula. The change of base formula allows us to convert to any base, but base 10 or base e (natural logarithm) are most convenient:

log₃(20) = log(20) / log(3) (using base 10)

or

log₃(20) = ln(20) / ln(3) (using base e)

Step 2: Evaluate. Now, use your calculator to evaluate the logarithms. Using base 10:

log₃(20) = log(20) / log(3) ≈ 1.3010 / 0.4771 ≈ 2.7268

Using base e:

log₃(20) = ln(20) / ln(3) ≈ 2.9957 / 1.0986 ≈ 2.7268

Either way, we find that log₃(20) ≈ 2.7268.

Encouraging Active Learning

The best way to learn logarithms is to practice!

Before looking at the solutions, actively try solving these examples yourself. Experiment with different bases and see how the properties apply. Remember, understanding why each step is taken is just as important as getting the correct answer.

So, there you have it! Hopefully, this step-by-step guide has made learning how to evaluate log expressions a little less intimidating. Keep practicing, and soon you'll be solving these problems with ease! Good luck!